A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12

Offset: 0

Author

Alois P. Heinz, Jun 06 2015

Keywords

• nonn
• tabl
• look

Examples

			Square array A(n,k) begins:
  0,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
  1,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
  2,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  3,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  4,  4,  4,  4,  4,   4,   4,   4,    4,    4, ...
  5,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  6,  5,  1,  0,  0,   0,   0,   0,    0,    0, ...
  7,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ...
  9,  6,  5,  1,  0,   0,   0,   0,    0,    0, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..115, flattened
• J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8
• Wikipedia, Arithmetic derivative

Crossrefs

Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.

Rows n=0,1,4,8 give: A000004, A000007, A010709, A129150.

Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.

Main diagonal gives A185232.

Antidiagonal sums give A258652.

Cf. also A328383.

Programs

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
seq(seq(A(n, h-n), n=0..h), h=0..14);

Mathematica

d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)

Formula

A(n,k) = A003415^k(n).